25.10: Appendix 25C Analytic Geometric Properties of Ellipses
- Page ID
- 25598
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Consider Equation (25.3.20), and for now take \(\varepsilon<1\), so that the equation is that of an ellipse. We shall now show that we can write it as
\[\frac{\left(x-x_{0}\right)^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \nonumber \]
where the ellipse has axes parallel to the x - and y -coordinate axes, center at (\(x_{0}\), 0) , semi-major axis a and semi-minor axis b . We begin by rewriting Equation (25.3.20) as
\[x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}} \nonumber \]
We next complete the square,
\[\begin{array}{l}
x^{2}-\frac{2 \varepsilon r_{0}}{1-\varepsilon^{2}} x+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{1-\varepsilon^{2}}+\frac{\varepsilon^{2} r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}+\frac{y^{2}}{1-\varepsilon^{2}}=\frac{r_{0}^{2}}{\left(1-\varepsilon^{2}\right)^{2}} \Rightarrow \\
\frac{\left(x-\frac{\varepsilon r_{0}}{1-\varepsilon^{2}}\right)^{2}}{\left(r_{0} /\left(1-\varepsilon^{2}\right)\right)^{2}}+\frac{y^{2}}{\left(r_{0} / \sqrt{1-\varepsilon^{2}}\right)^{2}}=1
\end{array} \nonumber \]
The last expression in (25.C.3) is the equation of an ellipse with semi-major axis
\[a=\frac{r_{0}}{1-\varepsilon^{2}} \nonumber \]
semi-minor axis
\[b=\frac{r_{0}}{\sqrt{1-\varepsilon^{2}}}=a \sqrt{1-\varepsilon^{2}} \nonumber \]
and center at
\[x_{0}=\frac{\varepsilon r_{0}}{\left(1-\varepsilon^{2}\right)}=\varepsilon a \nonumber \]
as found in Equation (25.B.10).